Question: Add the following rational expressions. $\dfrac{9y}{4y+5}+\dfrac{8y^2}{6y+5}=$
Answer: We can add two rational expressions whose denominators are equal by adding the numerators and keeping the denominator the same. [Does this fit with how we add rational numbers?] When the denominators are not the same, we must manipulate them so that they become the same. In other words, we must find a common denominator. Since the two denominators do not share any common factors, the common denominator is simply the product of these two denominators: $({4y+5})\cdot({6y+5})$. Let's manipulate the expressions to have that denominator: $\begin{aligned} &\phantom{=}\dfrac{9y}{{4y+5}}+\dfrac{8y^2}{{6y+5}} \\\\ &=\dfrac{9y\cdot({6y+5})}{({4y+5})\cdot({6y+5})}+\dfrac{8y^2\cdot({4y+5})}{({6y+5})\cdot({4y+5})} \end{aligned}$ [Why did we do that?] Now that both denominators are the same, let's add! $\begin{aligned} &\phantom{=}\dfrac{9y\cdot(6y+5)}{(4y+5)\cdot(6y+5)}+\dfrac{8y^2\cdot(4y+5)}{(6y+5)\cdot(4y+5)} \\\\ &=\dfrac{9y\cdot(6y+5)+8y^2\cdot(4y+5)}{(4y+5)(6y+5)} \\\\ &=\dfrac{54y^2+45y+32y^3+40y^2}{(4y+5)(6y+5)} \\\\ &=\dfrac{32y^3+94y^2+45y}{(4y+5)(6y+5)} \end{aligned}$ In conclusion, $\dfrac{9y}{4y+5}+\dfrac{8y^2}{6y+5}=\dfrac{32y^3+94y^2+45y}{(4y+5)(6y+5)}$